CN107203663A - Compliant member points to acquisition methods under a kind of motor-driven effect of rail control - Google Patents

Compliant member points to acquisition methods under a kind of motor-driven effect of rail control Download PDF

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CN107203663A
CN107203663A CN201710347072.7A CN201710347072A CN107203663A CN 107203663 A CN107203663 A CN 107203663A CN 201710347072 A CN201710347072 A CN 201710347072A CN 107203663 A CN107203663 A CN 107203663A
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CN107203663B (en
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邓润然
王大轶
邹元杰
葛东明
于登云
刘绍奎
朱卫红
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Beijing Institute of Spacecraft System Engineering
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Beijing Institute of Spacecraft System Engineering
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/15—Vehicle, aircraft or watercraft design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]

Abstract

Compliant member points to acquisition methods under a kind of motor-driven effect of rail control, carries out limited configurations Meta Model to reflector antenna first, and then sets up satellite Rigid-flexible Coupling Dynamics model, obtains reflector antenna mode of oscillation vector；Then set up satellite gravity anomaly action model, so with dynamical model under satellite Rigid-flexible Coupling Dynamics model composition control ring closure；Finally, expression formula of the aerial radiation field strength under antenna mode of oscillation space is set up, the in-orbit state kinetics attitude control aerial radiation collective model of whole star system is set up；The time-varying modal coordinate for obtaining satellite antenna vibration is emulated according to satellite operation on orbit excited data, radiation field of aerial Modal Space expression formula is substituted into, you can the dynamic change situation that Operational modes influence on aerial radiation field strength is obtained.

Description

Compliant member points to acquisition methods under a kind of motor-driven effect of rail control
Technical field
Compliant member points to acquisition methods under a kind of motor-driven effect of rail control of the present invention, is reflected for Large Deployable rope net Surface antenna operation on orbit process, mutation analysis technology neck is pointed to for antenna caused by day linearly coupled caused by satellite operation on orbit Domain.
Background technology
With the continuous improvement of spacecraft mission requirements, the complexity of spacecraft constantly rises, increasing big Type compliant member all realizes in-orbit application on spacecraft.In recent years, it is in as the satellite antenna of the important payload of satellite Point to high accuracy and the development trend of structure large-scale.Spaceborne heavy caliber reflector antenna belongs to typical large-scale flexible part, China is also developing all kinds of new spacecrafts with large-scale reflector antenna.This kind of spacecraft belong to big flexibility, low frequency, The complex dynamical systems of underdamping, while again to antenna beam pointing accuracy and stability, the requirement of type face precision index very Strictly.However, the Working mould such as position guarantor, attitude maneuver, solar wing driving, Yaw steering in satellite operation on orbit imaging process Formula, it is extremely sensitive to large-scale flexible antenna Imaging.Large-scale reflector antenna will be caused to vibrate, and then influence day line imaging The indexs such as the beam position stability of period, reduce satellite imagery quality.Large-scale reflector antenna working frequency range is higher, reflecting surface Radiance influence of the type facial disfigurement on antenna is larger, will by the beam position of Large Deployable film or cable mesh reflector antenna Ask, the structure design and technique to antenna propose higher requirement.High structure precision requirement can improve the service behaviour of antenna, But it can also greatly improve cost.Influenceed accordingly, it would be desirable to which research structure vibration is pointed on antenna beam, find influence pointing accuracy Principal element, and then can just provide the raising rational solution of pointing accuracy.Therefore, realize flexible under the motor-driven effect of rail control It is the basis for improving spaceborne large-scale reflector antenna pointing accuracy that part, which points to acquisition methods,.
At present, the conventional method for antenna beam direction analysis is generally the beam position point under static day line style face Analysis.And for satellite antenna, platform be under free boundary, while satellite with attitude control system act on, star body Motion necessarily causes the structural vibration of large-scale reflector antenna, and antenna structure is time-varying and non-static, the type facial disfigurement of antenna Moment changes, and its beam position also changes constantly.To this platform free floating, antenna structure vibration processes Beam position is analyzed, and the antenna of whole vibration processes can not be provided only with the beam position analysis under static day line style face Beam position change, it is impossible to efficiently provide worst antenna beam error in pointing, obviously can not so meet engineering need Ask.The beam position analysis of antenna should be carried out under whole star system, provide structural vibration to the direct relation of beam position, real The direction analysis of existing antenna time domain vibration processes.
The content of the invention
Present invention solves the technical problem that being：Overcoming the deficiencies in the prior art, there is provided under a kind of motor-driven effect of rail control Compliant member point to acquisition methods.By whole star dynamics, gesture stability, beam position collective model, in whole star system Level aspect, realizes the beam position analysis of the lower antenna structure vibration time-varying process of rail control effect.
The technical scheme that the present invention is solved is：Compliant member points to acquisition methods, step under a kind of motor-driven effect of rail control It is as follows：
(1) whole star Rigid-flexible Coupling Dynamics model is set up.
Wherein (1) formula be system barycenter translational motion equation, (2) formula be system around barycenter rotational motion equation, (3), (4) it is solar wing governing equation, (5), (6) are solar wing vibration equation, and (7) are antenna vibration equation.In formula：
ωsFor the angular speed array of satellite hub body；
For the antisymmetric matrix of angular speed array；
M is satellite Mass matrix；
IsFor satellite inertia battle array；
PsTo act on the external force array on satellite；
TsTo act on the moment of face array on satellite；
ωals、ωarsThe respectively angular speed array of left and right solar wing；
Ωals、Ωars、ΩaThe respectively modal frequency diagonal matrix of left and right solar wing and antenna；
ηls、ηrs、ηaRespectively left and right solar wing and antenna modal coordinate battle array；
ζls、ζrs、ζaRespectively the modal damping coefficient of left and right solar wing and antenna, typically takes 0.005；
Ials、IarsRespectively left and right solar wing inertia battle array；
Ftls、Ftrs、FtaThe respectively flexible couplings factor arrays of left and right solar wing and day linearly coupled to body translation；
Fsls、Fsrs、FsaThe flexible couplings factor arrays that respectively left and right solar wing and day linearly coupled are rotated to body；
Fals、FarsRespectively left and right solar wing vibrates the flexible couplings factor arrays to own rotation；
Rasls、RasrsRespectively left and right solar wing rotates the rigid coefficient of coup battle array rotated with satellite；
Tals、TarsThe control moment array respectively acted on the solar wing of left and right.
(2) attitude control of satellite simulation is set up.
In the case of a width of known conditions of gesture stability band of satellite, structural notch filter is put aside, proportional-plus-derivative control is determined System rule is as follows：
U=Kpθs+Kdωs (8)
Wherein KpFor proportional gain, KdFor the differential gain, θsFor whole star attitude angle, u is control moment.
The control-moment gyro on satellite is designed with momenttum wheel transmission function by as follows：
S is Laplace operator, ξsFor the damped coefficient of control-moment gyro.
S is Laplace operator, ξtFor the damped coefficient of momenttum wheel.
To sum up, the output control torque such as formula (11) realized by control-moment gyro and momenttum wheel,
Ts=Gt(s)Gs(s)(Kpθs+Kdωs) (11)
Whole star system under above-mentioned whole star Rigid-flexible Coupling Dynamics model, Attitude control model composition control ring closure Kinetic model：
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (19)
(3) the radiation field of aerial relation of structural vibration deformation and satellite in satellite operation on orbit is set up
According to physical optical method, the mirror surface induced-current in irradiated regionIt is expressed as
Wherein,For the position vector at arbitrfary point on the mirror surface of reflector antenna,For the anti-of reflector antenna Penetrate face surfaceThe outer normal vector of the unit at place,For the mirror surface of reflector antennaThe incident magnetic at place.
Obtain surface induction electric currentAfterwards, far-field approximation is introduced, then the radiated electric field produced by surface induction electric currentFor
Wherein, j is complex unit, and k is free-space propagation constant, and η is wave impedance, and r is distance of the point of observation to origin,For unit dyad,For unit vectorDyad, s be mirror surface area.By antenna in assigned direction radiation intensity The directivity factor of antenna can be obtained with the ratio between mean radiation intensity.
Radiation field of aerial is relevant with the change in location of reflecting surface arbitrfary point, and analysis is used as using reflector antenna FEM model Object, polynary Taylor expansion is used to formula (21) by each node location change of antenna reflective face FEM model：
In formula, q=[qx,qy,qz] it is the position on the mirror surface of reflector antenna at arbitrfary point along x, y, z three The projection scalar in direction；[qx0,qy0,qz0] for the initial position of arbitrfary point on the surface of antenna vibration deformation front-reflection face.
Antenna reflective face (i.e. type face) any point deformation is converted using modal coordinate：
Δ q=[Δ qx,Δqy,Δqz]=[φx,φy,φz]η (23)
In formula, [φx,φy,φz] it is translation mode of the arbitrfary point along three directions vibrations of x, y, z, η on mirror surface For mode of oscillation coordinate.
Assuming that taking Two-order approximation precision to meet demand the field strength E of radiation field of aerial, the mode of antenna radiation performance is sat Marking expression formula is：
E=E0+W1η+ηTW2η (24)
In formula, E0For the radiated electric field of initial time antenna before vibration.
Definition m is rank number of mode, and each variable expression is as follows in formula (16)：
E0=E (qx0,qy0,qz0) (25)
W1=[w1,w2,…wm] (26)
η=[η1,η2,…,ηm]T (29)
In formula, m is rank number of mode；[φi,x,φi,y,φi,z] be along three directions of x, y, z the i-th rank translation mode；
By with up conversion by aerial radiation electric field be the expression formula that is transformed under antenna mode of oscillation space of (22) formula i.e. Formula (24)~formula (29), according to the whole star Rigid-flexible Coupling Dynamics model of step (1), step (2) attitude control of satellite simulation and step Suddenly the Modal Space expression formula of the antenna radiation performance of (3), sets up the in-orbit state kinetics-attitude control-aerial radiation of whole star system Collective model：
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (37)
E=E0+W1η+ηTW2η (38)
(4) with the in-orbit state kinetics-attitude control-aerial radiation collective model of whole star system, ask for E to obtain a day linearly coupled During radiation field intensity time domain change, can be obtained in the ratio between assigned direction radiation field intensity and mean radiation intensity by antenna The directivity factor of antenna, that is, obtain antenna beam and point to.
The advantage of the present invention compared with prior art is：
(1) the in-orbit dynamics of the satellite with compliant member and gesture stability, electrical property comprehensive modeling are given, emulation, is divided Analysis method, the analysis method for providing complete set is pointed to for the flexible reflector antenna during satellite operation on orbit.
(2) by the system integration model of foregoing foundation, the modal coordinate of the day linearly coupled of time-varying is obtained, by determining day A constant not changed over time in beta radiation approximate expression, sets up antenna vibration processes radiance Two-order approximation mode empty Between expression formula.
(3) passed through it is radiation field of aerial with reflecting surface vibration deformation Two-order approximation Modal Space relational expression, during substitution The antenna mode of oscillation coordinate of change, it is possible to achieve the dynamic electrical performance analysis of antenna vibration processes under the in-orbit free state of satellite.
Brief description of the drawings
Fig. 1 flow charts of the method for the present invention；
Embodiment
The present invention basic ideas be：Compliant member points to acquisition methods under a kind of motor-driven effect of rail control, first to anti- Penetrate surface antenna and carry out limited configurations Meta Model, and then set up satellite Rigid-flexible Coupling Dynamics model, obtain reflector antenna vibration Modal vector；Then satellite gravity anomaly action model is set up, and then is closed with satellite Rigid-flexible Coupling Dynamics model composition control The lower dynamical model of ring effect；Finally, expression formula of the aerial radiation field strength under antenna mode of oscillation space is set up, is set up The whole in-orbit state kinetics-attitude control-aerial radiation collective model of star system；Obtained according to the emulation of satellite operation on orbit excited data The time-varying modal coordinate of satellite antenna vibration, substitutes into radiation field of aerial Modal Space expression formula, you can obtain Operational modes The dynamic change situation influenceed on aerial radiation field strength.
The present invention is described in further detail with specific embodiment below in conjunction with the accompanying drawings,
As shown in figure 1, compliant member points to acquisition methods under a kind of motor-driven effect of rail control of the present invention, step is as follows：
(1) whole star Rigid-flexible Coupling Dynamics model is set up.
Wherein (1) formula be system barycenter translational motion equation, (2) formula be system around barycenter rotational motion equation, (3), (4) it is solar wing governing equation, (5), (6) are solar wing vibration equation, and (7) are antenna vibration equation.In formula：
ωsFor the angular speed array of satellite hub body；
For the antisymmetric matrix of angular speed array；
M is satellite Mass matrix；
IsFor satellite inertia battle array；
PsTo act on the external force array on satellite；
TsTo act on the moment of face array on satellite；
ωals、ωarsThe respectively angular speed array of left and right solar wing；
Ωals、Ωars、ΩaThe respectively modal frequency diagonal matrix of left and right solar wing and antenna；
ηls、ηrs、ηaRespectively left and right solar wing and antenna modal coordinate battle array；
ζls、ζrs、ζaRespectively the modal damping coefficient of left and right solar wing and antenna, typically takes 0.005；
Ials、IarsRespectively left and right solar wing inertia battle array；
Ftls、Ftrs、FtaThe respectively flexible couplings factor arrays of left and right solar wing and day linearly coupled to body translation；
Fsls、Fsrs、FsaThe flexible couplings factor arrays that respectively left and right solar wing and day linearly coupled are rotated to body；
Fals、FarsRespectively left and right solar wing vibrates the flexible couplings factor arrays to own rotation；
Rasls、RasrsRespectively left and right solar wing rotates the rigid coefficient of coup battle array rotated with satellite；
Tals、TarsThe control moment array respectively acted on the solar wing of left and right.
(2) attitude control of satellite simulation is set up.
In the case of a width of known conditions of gesture stability band of satellite, structural notch filter is put aside, proportional-plus-derivative control is determined System rule is as follows：
U=Kpθs+Kdωs (8)
Wherein KpFor proportional gain, KdFor the differential gain, θsFor whole star attitude angle, u is control moment.
The control-moment gyro on satellite is designed with momenttum wheel transmission function by as follows：
S is Laplace operator, ξsFor the damped coefficient of control-moment gyro.
S is Laplace operator, ξtFor the damped coefficient of momenttum wheel.
To sum up, the output control torque such as formula (11) realized by control-moment gyro and momenttum wheel,
Ts=Gt(s)Gs(s)(Kpθs+Kdωs) (11)
Whole star system under above-mentioned whole star Rigid-flexible Coupling Dynamics model, Attitude control model composition control ring closure Kinetic model：
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (19)
(3) the radiation field of aerial relation of structural vibration deformation and satellite in satellite operation on orbit is set up
According to physical optical method, the mirror surface induced-current in irradiated regionIt is expressed as
Wherein,For the position vector at arbitrfary point on the mirror surface of reflector antenna,For the anti-of reflector antenna Penetrate face surfaceThe outer normal vector of the unit at place,For the mirror surface of reflector antennaThe incident magnetic at place.
Obtain surface induction electric currentAfterwards, far-field approximation is introduced, then the radiated electric field produced by surface induction electric currentFor
Wherein, j is complex unit, and k is free-space propagation constant, and η is wave impedance, and r is distance of the point of observation to origin,For unit dyad,For unit vectorDyad, s be mirror surface area.By antenna in assigned direction radiation intensity The directivity factor of antenna can be obtained with the ratio between mean radiation intensity.
Radiation field of aerial is relevant with the change in location of reflecting surface arbitrfary point, and analysis is used as using reflector antenna FEM model Object, polynary Taylor expansion is used to formula (21) by each node location change of antenna reflective face FEM model：
In formula, q=[qx,qy,qz] it is the position on the mirror surface of reflector antenna at arbitrfary point along x, y, z three The projection scalar in direction；[qx0,qy0,qz0] for the initial position of arbitrfary point on the surface of antenna vibration deformation front-reflection face.
Antenna reflective face (i.e. type face) any point deformation is converted using modal coordinate：
Δ q=[Δ qx,Δqy,Δqz]=[φx,φy,φz]η (23)
[φx,φy,φz] it is translation mode of the arbitrfary point along three directions vibrations of x, y, z on mirror surface, η is vibration Modal coordinate.
Assuming that taking Two-order approximation precision to meet demand the field strength E of radiation field of aerial, the mode of antenna radiation performance is sat Marking expression formula is：
E=E0+W1η+ηTW2η (24)
In formula, E0For the radiated electric field of initial time antenna before vibration.
Definition m is rank number of mode, and each variable expression is as follows in formula (16)：
E0=E (qx0,qy0,qz0) (25)
W1=[w1,w2,…wm] (26)
η=[η1,η2,…,ηm]T (29)
In formula, m is rank number of mode；[φi,x,φi,y,φi,z] be along three directions of x, y, z the i-th rank translation mode；
By with up conversion by aerial radiation electric field be the expression formula that is transformed under antenna mode of oscillation space of (22) formula i.e. Formula (24)~formula (29), according to the whole star Rigid-flexible Coupling Dynamics model of step (1), step (2) attitude control of satellite simulation and step Suddenly the Modal Space expression formula of the antenna radiation performance of (3), sets up the in-orbit state kinetics-attitude control-aerial radiation of whole star system Collective model：
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (37)
E=E0+W1η+ηTW2η (38)
(4) with the in-orbit state kinetics-attitude control-aerial radiation collective model of whole star system, ask for E to obtain a day linearly coupled During radiation field intensity time domain change, can be obtained in the ratio between assigned direction radiation field intensity and mean radiation intensity by antenna The directivity factor of antenna, that is, obtain antenna beam and point to.Analyzed with reference to concrete engineering, illustrate the present invention in application process In specific implementation step.
It is preferred that scheme be：Analysis pair is used as by the satellite of flexible accessory of two pieces of solar wings of band and large-scale reflector antenna As, it is assumed that solar wing is in the case of not driving, to analyze antenna principal direction radiance；
The first step, sets up antenna FEM model, obtains each rank mode of oscillation of antenna, and intercepting criterion by mode intercepts m The effective mode of rank；Set up solar wing FEM model；
Second step, determines the scalar matrix E in formula (38)0；
The reflector antenna not deformed calculated to obtain radiation field of aerial strong by engineering software；
3rd step, determines the constant W in formula (38)1；
Multiply the deformation of positive and negative unit modal coordinate amplitude for antenna the i-th rank mode respectively, two are calculated by engineering software It is E to plant deformation static antennas radiation fieldi, E-i；By Ei, E-iDeformation with positive and negative unit modal coordinate amplitude substitutes into foregoing The Two-order approximation modal coordinate expression formula of the aerial radiation field strength of foundation：
E=E0+W1η+ηTW2η (39)
Following formula can so be obtained：
Simultaneous above formula can be obtained：
Obtain constant：W1=[w1,w2,…wm]。
Similarly for appointing first-order modal to obtain above constant.
4th step, determines the constant W in formula (38)2；
Known foregoing derivation gives constant W2Expression formula it is as follows：
For W2In diagonal coupling terms wiwi, can be solved by formula (40), (41) summation simultaneous：
wiwi=Ei+E-i-2E0 (44)
For W2In off-diagonal coupling terms wiwj, add the positive unit modal coordinate of j rank mode to shake the i of antenna respectively Amplitude variation shape, antenna static radiation E is calculated by engineering softwareij.The positive unit modal coordinate that the i of antenna adds j rank mode is shaken The deformation of width substitutes into formula (39) and can obtained：
Again by the formula (42) of foregoing solution, (44), and the w similarly tried to achieve with formula (42), (44)j、wjwjSubstitute into above formula, It can solve：
wiwj=Eij-Ei-Ej+E0 (46)
Obtain by diagonal coupling terms wiwiWith off-diagonal coupling terms wiwjThe constant W of composition2。
So far, the constant E under the limited mode of antenna interception can be obtained0、W1、W2。
4th step, obtains antenna, the flexible vibration of two solar wings of left and right to the flexible couplings coefficient matrix of whole star translation： Fta、Ftrs、Ftls；The flexible couplings coefficient matrix that acquisition antenna, the flexible vibration of two solar wings of left and right are rotated to whole star：Fsa、 Fsls、Fsrs；
5th step, using Rigid Base band flexible accessory method, sets up the system dynamics side of the in-orbit free state of whole star Journey：
6th step, sets up attitude control of satellite simulation, and then constitute whole star dynamics-control-aerial radiation collective model；
Do not consider structural notch filter, determine proportional-plus-derivative control law, it is as follows：
U=Kpθs+Kdωs (52)
Wherein KpFor proportional gain, KdFor the differential gain, θsFor whole star attitude angle, u is control moment.
Design the control-moment gyro and momenttum wheel transmission function G on satellites(s), momenttum wheel transmission function Gt(s), by controlling Moment gyro processed and the output control torque that momenttum wheel is realized are as follows：
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (53)
So far, whole star dynamics-control-aerial radiation collective model is constituted：
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (59)
7th step, adds the in-orbit each rail control mode of operation of satellite, and solve each reservation mode of correspondence changes over time mould State coordinate, substitutes into the radiation field that formula (60) solves each moment reflector antenna principal direction.
The present invention has passed through radiation field of aerial with reflecting surface vibration deformation Two-order approximation Modal Space relational expression, substitutes into The antenna mode of oscillation coordinate of time-varying, it is possible to achieve the dynamic electrical performance of antenna vibration processes point under the in-orbit free state of satellite Analysis.

Claims (4)

1. compliant member points to acquisition methods under a kind of motor-driven effect of rail control, it is characterised in that step is as follows：
(1) whole star Rigid-flexible Coupling Dynamics model is set up；
(2) according to whole star Rigid-flexible Coupling Dynamics model, attitude control of satellite simulation, whole star system power under gesture stability is set up Learn model；
(3) according to the Modal Space table of whole star Rigid-flexible Coupling Dynamics model, attitude control of satellite simulation and antenna radiation performance Up to formula, the in-orbit state kinetics-attitude control-aerial radiation collective model of whole star system is set up；
(4) with the in-orbit state kinetics-attitude control-aerial radiation collective model of whole star system, obtain in antenna vibration processes and radiate The time domain change of field strength, the direction of antenna can be obtained by antenna in the ratio between assigned direction radiation field intensity and mean radiation intensity Property coefficient, that is, obtain antenna beam and point to.
2. compliant member points to acquisition methods under the motor-driven effect of a kind of rail control according to claim 1, it is characterised in that： The whole star Rigid-flexible Coupling Dynamics model that step (1) is set up, it is as follows：
<mrow> <mi>M</mi> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mi>s</mi> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msub> <mover> <mi>&amp;omega;</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <msub> <mi>I</mi> <mi>s</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mi>a</mi> </msub> <msub> <mi>&amp;Omega;</mi> <mi>a</mi> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mi>a</mi> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mi>a</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Wherein (1) formula be system barycenter translational motion equation, (2) formula be system around the rotational motion equation of barycenter, (3), (4) are Solar wing governing equation, (5), (6) are solar wing vibration equation, and (7) are antenna vibration equation, in formula：
ωsFor the angular speed array of satellite hub body；
For the antisymmetric matrix of angular speed array；
M is satellite Mass matrix；
IsFor satellite inertia battle array；
PsTo act on the external force array on satellite；
TsTo act on the moment of face array on satellite；
ωals、ωarsThe respectively angular speed array of left and right solar wing；
Ωals、Ωars、ΩaThe respectively modal frequency diagonal matrix of left and right solar wing and antenna；
ηls、ηrs、ηaRespectively left and right solar wing and antenna modal coordinate battle array；
ζls、ζrs、ζaRespectively the modal damping coefficient of left and right solar wing and antenna, typically takes 0.005；
Ials、IarsRespectively left and right solar wing inertia battle array；
Ftls、Ftrs、FtaThe respectively flexible couplings factor arrays of left and right solar wing and day linearly coupled to body translation；
Fsls、Fsrs、FsaThe flexible couplings factor arrays that respectively left and right solar wing and day linearly coupled are rotated to body；
Fals、FarsRespectively left and right solar wing vibrates the flexible couplings factor arrays to own rotation；
Rasls、RasrsRespectively left and right solar wing rotates the rigid coefficient of coup battle array rotated with satellite；
Tals、TarsThe control moment array respectively acted on the solar wing of left and right.
3. compliant member points to acquisition methods under the motor-driven effect of a kind of rail control according to claim 1, it is characterised in that： Step (2) sets up whole star system power under gesture stability according to whole star Rigid-flexible Coupling Dynamics model, attitude control of satellite simulation Model is learned, method is as follows：
In the case of a width of known conditions of gesture stability band of satellite, structural notch filter is put aside, proportional-plus-derivative control is determined Rule is as follows：
U=Kpθs+Kdωs (8)
In formula, KpFor proportional gain, KdFor the differential gain, θsFor whole star attitude angle, u is control moment, the control on design satellite Moment gyro is with momenttum wheel transmission function by as follows：
<mrow> <msub> <mi>G</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msubsup> <mi>&amp;omega;</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;xi;</mi> <mi>s</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>s</mi> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>&amp;omega;</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
S is Laplace operator, ξsFor the damped coefficient of control-moment gyro；
<mrow> <msub> <mi>G</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msubsup> <mi>&amp;omega;</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;xi;</mi> <mi>t</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>t</mi> </msub> <mi>s</mi> <mo>+</mo> <msubsup> <mi>&amp;omega;</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
S is Laplace operator, ξtFor the damped coefficient of momenttum wheel；
By control-moment gyro and momenttum wheel, the output control torque such as formula (11) of realization,
Ts=Gt(s)Gs(s)(Kpθs+Kdωs) (11)
Whole star system power under above-mentioned whole star Rigid-flexible Coupling Dynamics model, Attitude control model composition control ring closure Learn whole star system kinetic model under model, i.e. gesture stability：
<mrow> <mi>M</mi> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mi>s</mi> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msub> <mover> <mi>&amp;omega;</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <msub> <mi>I</mi> <mi>s</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mi>a</mi> </msub> <msub> <mi>&amp;Omega;</mi> <mi>a</mi> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mi>a</mi> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mi>a</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (19)。
4. compliant member points to acquisition methods under the motor-driven effect of a kind of rail control according to claim 1, it is characterised in that： Step (3) is according to the Modal Space table of whole star Rigid-flexible Coupling Dynamics model, attitude control of satellite simulation and antenna radiation performance Up to formula, the in-orbit state kinetics-attitude control-aerial radiation collective model of whole star system is set up, method is as follows；
According to physical optical method, the mirror surface induced-current in irradiated regionIt is expressed as
<mrow> <mover> <mi>J</mi> <mo>&amp;RightArrow;</mo> </mover> <mrow> <mo>(</mo> <mover> <mi>q</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mover> <mi>n</mi> <mo>^</mo> </mover> <mo>&amp;times;</mo> <mover> <mi>H</mi> <mo>&amp;RightArrow;</mo> </mover> <mrow> <mo>(</mo> <mover> <mi>q</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
Wherein,For the position vector at arbitrfary point on the mirror surface of reflector antenna,For the reflecting surface of reflector antenna SurfaceThe outer normal vector of the unit at place,For the mirror surface of reflector antennaThe incident magnetic at place；
Obtain surface induction electric currentAfterwards, far-field approximation is introduced, then the radiated electric field produced by surface induction electric currentFor
<mrow> <mover> <mi>E</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>j</mi> <mi>k</mi> <mi>&amp;eta;</mi> <mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mi>k</mi> <mi>r</mi> </mrow> </msup> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> <mi>r</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mover> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;OverBar;</mo> </mover> <mo>-</mo> <mover> <mi>r</mi> <mo>^</mo> </mover> <mover> <mi>r</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <munder> <mo>&amp;Integral;</mo> <mi>s</mi> </munder> <mover> <mi>J</mi> <mo>&amp;RightArrow;</mo> </mover> <mrow> <mo>(</mo> <mover> <mi>q</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mi>k</mi> <mover> <mi>q</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;CenterDot;</mo> <mover> <mi>r</mi> <mo>^</mo> </mover> </mrow> </msup> <mi>d</mi> <mi>s</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
Wherein, j is complex unit, and k is free-space propagation constant, and η is wave impedance, and r is distance of the point of observation to origin,For Unit dyad,For unit vectorDyad, s be mirror surface area；By antenna in assigned direction radiation intensity with putting down The ratio between equal radiation intensity can obtain the directivity factor of antenna；
Radiation field of aerial is relevant with the change in location of reflecting surface arbitrfary point, and analysis pair is used as using reflector antenna FEM model As using polynary Taylor expansion by each node location change of antenna reflective face FEM model to formula (21)：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>E</mi> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mo>!</mo> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>&amp;Delta;q</mi> <mi>x</mi> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;Delta;q</mi> <mi>y</mi> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;Delta;q</mi> <mi>z</mi> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msup> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>q</mi> <mrow> <mi>x</mi> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>y</mi> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>z</mi> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>&amp;Delta;q</mi> <mi>x</mi> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;Delta;q</mi> <mi>y</mi> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;Delta;q</mi> <mi>z</mi> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </msup> <mi>E</mi> <mo>(</mo> <msub> <mi>q</mi> <mrow> <mi>x</mi> <mn>0</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>&amp;Delta;q</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>y</mi> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>&amp;Delta;q</mi> <mi>y</mi> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>z</mi> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>&amp;Delta;q</mi> <mi>z</mi> </msub> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>&amp;theta;</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
In formula, q=[qx,qy,qz] it is the position on the mirror surface of reflector antenna at arbitrfary point along three directions of x, y, z Projection scalar；[qx0,qy0,qz0] for the initial position of arbitrfary point on the surface of antenna vibration deformation front-reflection face；
To antenna reflective face, i.e. type face, any point deformation is converted using modal coordinate：
Δ q=[Δ qx,Δqy,Δqz]=[φx,φy,φz]η (23)
In formula, [φx,φy,φz] it is translation mode of the arbitrfary point along three directions vibrations of x, y, z on mirror surface, η is to shake Dynamic modal coordinate；
Assuming that taking Two-order approximation precision to meet demand, the modal coordinate table of antenna radiation performance the field strength E of radiation field of aerial It is up to formula：
E=E0+W1η+ηTW2η (24)
In formula, E0For the radiated electric field of initial time antenna before vibration；
Definition m is rank number of mode, and each variable expression is as follows in formula (16)：
E0=E (qx0,qy0,qz0) (25)
W1=[w1,w2,…wm] (26)
<mrow> <msub> <mi>W</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>...</mo> <msub> <mi>w</mi> <mi>m</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>...</mo> <msub> <mi>w</mi> <mi>m</mi> </msub> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>&amp;phi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;phi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>y</mi> </mrow> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msub> <mi>&amp;phi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mfrac> <mo>&amp;part;</mo> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mi>x</mi> </msub> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mi>E</mi> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mrow> <mi>x</mi> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>y</mi> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>z</mi> <mn>0</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>...</mo> <mi>m</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>
η=[η1,η2,…,ηm]T (29)
In formula, m is rank number of mode；[φi,x,φi,y,φi,z] be along three directions of x, y, z the i-th rank translation mode；
By by aerial radiation electric field being expression formula i.e. formula that (22) formula is transformed under antenna mode of oscillation space with up conversion (24)~formula (29), according to the whole star Rigid-flexible Coupling Dynamics model of step (1), step (2) attitude control of satellite simulation and step (3) the Modal Space expression formula of antenna radiation performance, sets up the in-orbit state kinetics-attitude control-aerial radiation of whole star system comprehensive Matched moulds type,
<mrow> <mi>M</mi> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mi>s</mi> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msub> <mover> <mi>&amp;omega;</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <msub> <mi>I</mi> <mi>s</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow> 3
<mrow> <msub> <mi>I</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>I</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>l</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>Q</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>a</mi> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>&amp;zeta;</mi> <mi>a</mi> </msub> <msub> <mi>&amp;Omega;</mi> <mi>a</mi> </msub> <msub> <mover> <mi>&amp;eta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>a</mi> </msub> <mo>+</mo> <msubsup> <mi>&amp;Omega;</mi> <mi>a</mi> <mn>2</mn> </msubsup> <msub> <mi>&amp;eta;</mi> <mi>a</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> </mrow> <mi>T</mi> </msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>+</mo> <msubsup> <mi>F</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>
Tc=Gt(s)Gs(s)(Kpθs+Kdωs) (37)
E=E0+W1η+ηTW2η (38)。
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CN109032159A (en) * 2018-07-25 2018-12-18 中国空间技术研究院 A kind of whole star flexible dynamics model loading big flexible antenna determines method
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CN112327665B (en) * 2020-09-29 2024-05-10 北京空间飞行器总体设计部 Satellite large-scale assembly rigidity control method based on half-power bandwidth in multi-satellite transmission
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